Sounds like it could be a battery/alternator problem. The power steering not functioning could be a result of insufficient current being supplied to the system due to a faulty alternator/weak battery. That also explains why all the dashboard lights turned on at one point, as that happens to a lot of rav's when there is insufficient power.
As you specifically stated, this happens after a "cold start". This is usually when the battery is at it's weakest before the alternator charges it back up after engine start. I'd get the alternator and battery checked if it were my rav.